Quadrilaterals & Polygon Properties

The quadrilateral family is a ladder of conditions. Each rung adds one requirement and inherits everything above.

Here are six four-sided shapes. What do they have in common?

Four straight sides, joined up. That is the entire list.

But some of them have more. A parallelogram has both pairs of sides parallel — an extra condition.

A rectangle is a parallelogram that also has four right angles. A rhombus is one with four equal sides.

A square has both extra conditions, which is why it sits at the bottom of the family.

So a square is a rectangle, and it is also a rhombus, and a parallelogram, and a quadrilateral. Each rung inherits everything above it — and that is what makes "prove it is a rectangle" a finite job.

Angle sums

Any polygon with n sides splits into n - 2 triangles by drawing diagonals from one vertex.

So the interior angles sum to (n-2) × 180°. A quadrilateral gives 360°, a pentagon 540°, a hexagon 720°.

The exterior angles of any convex polygon always add to 360° — whatever n is. Walk once round the outside and you have turned through one full circle.

Regular polygons

In a regular n-gon each exterior angle is 360n and each interior angle is 180 - 360n. That is usually quicker than the (n-2)180 route.

What a parallelogram gives you

Opposite sides are congruent and parallel. Opposite angles are congruent. Consecutive angles are supplementary.

And the diagonals bisect each other — they cut each other exactly in half at their crossing point.

Each of these is also a test: prove any one of them and you have proved the shape is a parallelogram.

Rectangles, rhombuses, squares

A rectangle is a parallelogram with a right angle — and then all four are right angles. Its diagonals are congruent.

A rhombus is a parallelogram with two adjacent sides congruent — and then all four are. Its diagonals are perpendicular and bisect the angles.

A square is both, so it collects every property on the list: congruent diagonals that are also perpendicular.

Diagonals tell them apart

All parallelograms have diagonals that bisect. Only rectangles have congruent diagonals; only rhombuses have perpendicular ones. A square has both.

Trapezoids

A trapezoid has exactly one pair of parallel sides, called the bases. The other two are the legs.

An isosceles trapezoid has congruent legs, and then its base angles are congruent and its diagonals are congruent too.

The midsegment joins the midpoints of the legs and its length is the average of the two bases: b₁ + b₂2.

Proving which one it is

To prove a quadrilateral is a parallelogram, show any one of: both pairs of opposite sides parallel, both pairs congruent, one pair both parallel and congruent, or diagonals bisecting.

To upgrade to a rectangle, add a right angle or congruent diagonals. To a rhombus, add congruent adjacent sides or perpendicular diagonals.

Pick the test that matches what you were given — that is the whole skill.

Worked examples

Angle sum

Find the sum of the interior angles of a decagon, and the size of one interior angle if it is regular.

n = 10, so it splits into 10 - 2 = 8 triangles. Diagonals from one vertex.
Sum = 8 × 180 = 1440° The interior angle sum.
Regular, so divide: 1440 ÷ 10 = 144° Each interior angle.
Check via exteriors: 360 ÷ 10 = 36, and 180 - 36 = 144 ✓ The quicker route agrees.

1440°; each angle 144°

Parallelogram algebra

In parallelogram ABCD, ∠ A = (3x + 10)° and ∠ C = (5x - 30)°. Find ∠ B.

∠ A and ∠ C are opposite, so they are congruent. A parallelogram property.
3x + 10 = 5x - 30, so 40 = 2x and x = 20 Solve for x.
∠ A = 3(20) + 10 = 70° Substitute back.
∠ B is consecutive to ∠ A, so supplementary: 180 - 70 = 110° Not congruent — consecutive angles add to 180.

∠ B = 110°

Which quadrilateral?

A quadrilateral has diagonals that bisect each other and are congruent, but are not perpendicular. What is it?

Diagonals bisect each other, so it is a parallelogram. That is the parallelogram test.
Congruent diagonals upgrade it to a rectangle. Only rectangles have that.
Not perpendicular, so it is not a rhombus — and therefore not a square. A square would need both.
It is a rectangle that is not a square. Every condition accounted for.

A rectangle (not a square)

Words to know

Polygon: A closed figure made of straight segments that meet only at endpoints.
Regular polygon: All sides equal and all angles equal.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Rhombus: A parallelogram with four congruent sides.
Trapezoid: A quadrilateral with exactly one pair of parallel sides.
Diagonal: A segment joining two non-adjacent vertices.

Mistakes to watch for

Thinking a square is not a rectangle. It is — a square satisfies every rectangle condition and adds one more. The family is a ladder of inheritance, so the more specific shape always keeps the general one's properties.

Making consecutive angles in a parallelogram congruent. Opposite angles are congruent; consecutive ones are supplementary. They are only equal in the special case of a rectangle, where both are 90°.

Confusing which diagonals do what. All parallelograms have diagonals that bisect each other; congruent diagonals mean a rectangle, perpendicular ones mean a rhombus. Mixing these up names the wrong shape.

Using (n-2)180 for exterior angles. The exterior angles of any convex polygon always total 360°, no matter how many sides — you walk round once and turn through one full circle.

The word trapezoid means opposite things in different countries. In the US it is a quadrilateral with exactly one pair of parallel sides; in the UK that shape is a trapezium, and trapezoid means no parallel sides at all.

Both come from the Greek trapeza, "table" — a four-legged thing.

Exam boards on each side of the Atlantic have quietly accepted the clash for over a century.

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